\(a.\)
VOAs and Higgs branch geometry. It was shown in [1]} that there exists a map from (unitary) four-dimensional \({\cal N}=2\)  SCFTs to a restricted subset of (non-unitary) two-dimensional vertex operator algebras (VOAs).
This map identifies the Schur index of the \({\cal N}=2\)  theory with the vacuum character \(\chi _0\)  of the VOA, up to an overall factor,
\(\chi _0(q) = q^{- {c_{\mathrm {2d}}\over 24}}Z_{\rm Schur} (q) = q^{- {c_{\mathrm {2d}}\over 24}}(1 +\sum _{n\in {{\mathbb {N}}/ 2}} a_n q^n)~, \hspace{28.90755pt} a_n \in {\mathbb {Z}}~.\) 
Here \(c_{\mathrm {2d}}\)  is the 2d central charge, related to the 4d Weyl anomaly coefficient \(c_{4d}\)  by
\(c_{2d} = -12 c_{4d}\) . The vacuum character must have integral Fourier expansion (since it counts operators in the theory), as well as unit leading coefficient (signifying the presence of a unique vacuum).
Crucially, the VOAs associated to 4d SCFTs are believed to be of a special type, known as “quasi-lisse," which ensures that their vacuum characters satisfy a monic modular differential equations (MDE).A monic MDE is an MDE with unit leading coefficient and holomorphic coefficient functions. See Section REF  for explicit examples.
The modular properties of the vacuum character allow one to control its high temperature limit, from which one can extract the other Weyl anomaly coefficient \(a_{4d}\) . These facts will provide the starting point for our analysis. By looking for integral solutions to monic MDEs of fixed order, we can identify Schur indices and central charges of putative \({\cal N}=2\)  SCFTs.
The Schur sector of an \({\cal N}=2\)  SCFT comprises the Higgs branch chiral ring. In particular, the Fourier coefficent \(a_1\)  counts the number of moment maps of the 4d theory, i.e. the bottom components of the flavor current supermultiplets. In fact, according to the central conjecture of [2]}, the entire Higgs branch (as a complex sympletic variety) is encoded in the VOA.However, in this work we will only discuss putative vacuum characters and MDEs, which are often insufficient to reconstruct the full VOA.
\(b.\)
Coulomb branch geometry.
Conjecturally, every 4d \({\cal N}=2\)  theory has a branch of the moduli space with the property that the low-energy theory at a generic point is a free \({\cal N}=2\)  supersymmetric \(U(1)^r\)  gauge theory with no massless charged states. This is known as the Coulomb branch (CB). The quantity \(r\)  is called the rank of the theory and coincides with the complex dimensionality of the CB. The CB is singular,Purely on the basis of \({\cal N}=2\)  superconformal analysis, the CB can admit both metric and complex structure singularities. While the former arise when BPS states charged under the \(U(1)^r\)  become massless, the latter signal the fact that the CB chiral ring is not freely generated [3]}, [4]}. Throughout this manuscript we will restrict to \({\cal N}=2\)  theories with freely generated CB chiral ring and thus only consider metric singularities. and its (complex codimension-one) singular loci encode many properties of the theory. A related fact is that a globally-defined Lagrangian description of the low-energy \(U(1)^r\)  theory is only possible by allowing electro-magnetically dual changes of basis. This gives rise to non-trivial monodromies (lending the CB a Special Kähler structure [5]}, [6]}, [7]}), which constrain the properties of CB operators.
The CB monodromies, which take value in the discrete electro-magnetic duality group \(Sp(2r,\mathbb {Z})\) , can be used to restrict the scaling dimensions of CB operators to a very small set of rational values, which depends solely on the rank of the theory [8]}, [9]}. Furthermore, the stratification of the CB singular locus [10]}, [11]}, [12]} allows us to determine Weyl anomalies as well as flavor symmetry data in terms of the scaling dimension of CB observables.
The structure of the CB geometry thus constrains some of the same SCFT data—Weyl anomalies and flavor central charges—as are captured by the VOA. The CB and VOA constraints are largely independent, making their compatibility a highly non-trivial requirement.

